© 11' .\( 1 t'il'llIlI,d \\ "'l ld (,"lI),(!l '" BIIII"pt·,I. \. DYNAMIC STATE ESTIMATOR FOR LARGE ELECTRIC POWER SYSTEMS P. Rou sseaux , Th. Van Cutsem and M. Ribbens -Pavella Abstract. A method for performing dynamic state estimation in electric power systems is set forth. It deriv es from the conjunction of an extended Kalman filter and of a hierar - chical scheme, according to which the overall dynamic state estimation problem is de - composed into snaller, easier to handle subproblems. Aside from making the solution' of the large-scale problem at all feasible in practice, this leads to important computa- tional s avi ngs . Use d in real power systems, this structure succeeds indeed in keeping the advantages of th e Kalman filtering while clearing it from its extremely demanding com- puter time and stora ge requirements. A simple but realistic example il lustrates the method and shows its e ffectiveness. Kevwords. State estimation; power systems; Kalman filters; hierarchical systems; large- scale systems. NOTATION N.B. All symbols are fully defined at the place they are first introduced. As a convenience to the reader we have collected below the m ore used symbols in several places. (JJdh Mme exce ptioYL6 £owe Jt C(B e dauc £et- tvv., -<.niUca:t e and caDda.1 da£{.c £etieJtf.> denot e . k denotes a time sample . 6t = time interval. Z m - dimensional measurement vector. V measurement noise vector. R covariance matrix of the measurement noise vec - tor. a (j) : standard deviation of the j- th measurement noise. X : n - dimensional state vector. h (.) : nonlinear vector function relating measure - men t s Z to x ; h ( .) : -+ • W : system noise vector. Q : covariance matrix of the system noise vector. q (j) : maximum rate of change of the j - th state variable. P covariance matrix of the estimated state vector. K Kalman gain. b '" N[m, C] indicate s that vector b is a gaussian (normal) random vector with mean m and covari - ance matrix C. ISE : "Integrated" State Estimation, i.e. estimation of the system taken as a whole. HSE : "Hierarchical " State Estimation , i.e. estima - tion provided by a hierarchical scheme, through system decomposition . x estimated value of x provided through an ISE . x esti m ated value of x provided at the end of the first level of the HSE. x estimated value of x provided at the end of the second level of the HSE. 6 ij : Kronecker 's delta: 6 ij = 1 ( 0) if i =j (iF j) . E = expectation. diaa diagonal matrix. AT A transposed. INTRODUCTION During the past de c ade attempts were made to explore the time varying characteristics of the state of an 2 12 1 electric power system. They can be classified into two categories : tracking state estimation (SE) which extends techniques deve l oped for static SE without explicit definition of the dynamic model (Masiello and Schweppe, 197 0) and dynamic SE based on extended Kalman filtering (EKF) techniques. Within this latter approach, the design of an esti - mato r has to comply with specific r equirements , non - existent in the static one : the system mode ll ing has to be realistic , i.e . able to follow the time evolution and to detect sudden, unpredictable variations whatever their nature; but it has al so to remain algorithmically trackable so as to comp ly with on - line computer time and storage requirements. A priori, these two sets of conditions may appear rather contradictory and their satisfaction problem- atic. The first attempts towards dynamic SE (DSE) have quite naturally been concentrated on its very feasi- bility , essential ly by exploring possibility of de - riving valid dynamic models. A simple model has first been proposed by Debs and Larson (1970) and used within EKF techniques (Kalman, 1960; Kalmanand Bucy , 1961). Later on , this mode l was completed th r ough a " trend factor " (Nishiya et al. , 1976) and further imp r oved through an "innovation f actor" (Nishiya et al., 1980, 1982). The successive improvements suc - ceeded in building DSE algorithms capable of han - dling sudden variations. But at the same time a di - mensionality problem arose, and as a corrolary a computational burden, which moreove r amp l ifies very rapidly with the system size. And till now, the very few examples treated in the publications concern un- rea l istically small power systems. This paper addresses essential ly the latter problem , by setting forth dynamic algo r ithms able to conform to reliability and on -l ine requirements of rea l istic (i.e. large - scale) power systems. The proposed method consists in decomposing the overall system into sub - systems and in performing a m ultilevel state est i ma - tion. In its simplest version, the latter reduces to a two-level estimation , at the f irst leve l of which local SEs are performed independent ly for each subsystem; the coordination of these SEs is then rea l ized at the second level by an appropriate al go - rithm . The structure of this two - level procedure is inspired by the one developed and successfu ll y ap - plied for static SE purposes by Van Cutsem et a l. (1981) . To distinguish the algorithm derived here